I've got two Reilly .500 BPE's. One of them has developed a problem. The gun has rebounding hammers. The problem is with the right side. The hammer will not set at either half-cock or full-cock unless you lightly push the corresponding trigger forward until it can't move anymore.

What's happening and how serious is the fix?

Sounds like a broken sear spring.

That would be my first guess as well. Depends upon the design but 2-300$ will usually get you sorted.

Can you feel a spring pushing back when you pull each trigger without cocking the hammers? Could be a broken spring, or just a loose screw holding the trigger spring. Or crud in the way of the trigger/sear.

Send it back for warranty!